DIFFERENTIAL EQUATION: PARALLEL LINES

Show straightforward, complete, and neat solution. Box final answer.

\fDifferential Equation with Coefficients Linear in Two Variables: Parallel Lines EXAMPLE OF SOLUTION (need to see in the solution) Checking if Parallel or not Use u-substitution EX. Ex. Determine if the given DEs are parallel lines or not. 1. (x + y - 1) dy + (2x + 2y - 3) dx = 0 Check using the conditions: u = (A,x + Bly) OR B1 A1 A2 C2 = 1 (2) ;BB1 B2 (1): Az A1 C2 - 1 (3) u = (A2x + Bzy) - 3 - - 3 (2 ): 2 ( 3 ) du = (Ajdx + Bjdy) All conditions passed. OR Going back to the definition, we can conclude that the DE is parallel. du = (Azdx + Bzdy) #1: (x + 2y -3)dx - (2x + 4y - 1)dy = 0 Let u = (x + 2y); du = (dx + 2dy) - dx = (du - 2dy) (u - 3)(du - 2dy) - (2u - 1) dy = 0 (u - 3) du - (2u - 6) dy - (2u - 1) dy = 0 (u - 3) du - (4u - 7) dy = 0 (u - 3) du - (4u - 7) dy = 0 4u - 7 (u - 3) 4u - 7 -du - dy = 0 (u - 3) 7 du - dy = 0 - 4-4u-7 du - dy = 0 In(4u - 7) -y = C (x + 2y) -In(4 ( x+ 2y) - 7) - y = C 4(x + 2y) - 5 In(4x + 8y - 7) - 16y = C 4x - 8y - 5 In(4x + 8y - 7) = C - Final